Chaos and Phase-Insensitive Statistics

Summary

Why conventional likelihood-based inference collapses for chaotic dynamic models, and how to fix it. Because trajectories are hypersensitive to the noise realization and parameters, the joint density $f_{\theta }(\mathbf{y},\mathbf{e})$ of data and process noise is a wildly irregular, multimodal function of both $\mathbf{e}$ and $\mathbf{\theta }$ — useless for measuring fit and intractable to integrate or sample. The resolution is philosophical: the local phase of the data is a purely noise-driven, non-repeatable feature that should not enter any model–data comparison. So one judges fit using phase-insensitive summary statistics that capture the dynamically-important structure (autocovariances, autoregression coefficients) while discarding local phase.

Overview

This note develops the obstacle that motivates the synthetic likelihood and the design principle behind its summary statistics. The prototypical example is the scaled Ricker map, which shows the collapse of standard methods under chaotic dynamics.

Main Content

The scaled Ricker map (Wood 2010, Eq. 1)

The prototypic ecological model with complex dynamics describes a population $N_t$ by

$$N_{t+1}=r{N}_t{e}^{-N_t+e_t},$$

where $e_t\sim N(0,{\sigma }_e^2)$ are independent process-noise terms and $r$ is the intrinsic growth-rate parameter controlling the dynamics. Observations are Poisson deviates $y_t$ with mean $\varphi N_t$ (a common sampling situation). The inference target is ${\mathbf{\theta }}^{\top }=(r,{\sigma }_e^2,\varphi )$. For $\log r=3.8$ the dynamics are chaotic.

The likelihood collapse

To do likelihood-based inference one must integrate the joint density $f_{\theta }(\mathbf{y},\mathbf{e})$ over all process-noise vectors $\mathbf{e}$. But (Wood 2010, Fig. 1b–c):

• As a function of the noise $\mathbf{e}$: holding $\mathbf{y}$ and $\mathbf{\theta }$ fixed and varying a single noise deviate $e_1$, $\log f_{\theta }(\mathbf{y},\mathbf{e})$ is extraordinarily jagged and multimodal — so the integral over $\mathbf{e}$ is analytically and numerically intractable.
• As a function of $\mathbf{\theta }$: varying $r$ with $\mathbf{e},\mathbf{y}$ fixed makes $\log f_{\theta }$ equally irregular.

Bayesian inference fares no better: sampling $(\mathbf{e},\mathbf{\theta })$ from a density proportional to such an irregular $f_{\theta }$ has no workable method. Since this joint density underlies all conventional fit measures, the chaotic regime defeats them.

Why local phase must be discarded (Wood 2010, "Defining fit")

Naive inference tries to make the model reproduce the exact course of the observed data — something the real system itself would not do if repeated. The dynamic processes driving the system are a repeatable feature about which inference is legitimate, but the local phase of the data is an entirely noise-driven feature that should contribute nothing to a measure of match or mismatch. To be scientifically meaningful, statistical fit must be judged on quantities that reflect what is dynamically important, discarding local phase. (The idea of phase-insensitive comparison is not itself new; what is new is assessing the consistency of model-simulation and data statistics in a way that gives access to likelihood-based inference.)

Designing the summary statistics

The first analysis step reduces the raw data $\mathbf{y}$ to a vector of summary statistics $\mathbf{s}$ designed to capture the dynamic structure of the model — specifying what matters about the dynamics, but not how much it matters. Suitable examples:

• coefficients of the autocovariance function;
• coefficients of polynomial autoregressive models of the dynamics;
• coefficients from polynomial regression of the observed order statistics on fixed reference quantiles (to summarize the marginal distribution).

Using regression coefficients as statistics promotes approximate normality of $\mathbf{s}$, which supports the key multivariate-normal approximation $\mathbf{s}\sim N({\mathbf{\mu }}_{\theta },{\mathbf{\Sigma }}_{\theta })$ used in the Synthetic Likelihood Construction. There is complete freedom to transform statistics to improve this approximation, and the method is robust to including uninformative statistics, so careful selection is not essential.

Connections

• The obstacle motivates the Synthetic Likelihood Construction, which replaces the intractable $f_{\theta }$ with a smooth MVN likelihood of $\mathbf{s}$.
• Phase-insensitive summary statistics are the same device used in ABC and in Indirect Inference / Method of Simulated Moments (where an auxiliary model’s coefficients or moments serve as the statistics).
• Chaotic sensitivity to initial conditions is the hallmark of the dynamical systems studied; the scaled Ricker map recurs in ecology much as the Brock–Mirman map does in economics.

See Also

• Synthetic Likelihood - Overview — where this obstacle fits in the method
• Synthetic Likelihood Construction — the synthetic likelihood built on these statistics
• Nicholson’s Blowfly Application — the specific statistics used in practice